Using Cancellation to Add Positives and Negatives

Date: 10/27/2003 at 11:44:04
From: Ivette
Subject: algebra 

Why is algebra so hard? What can I do to memorize formulas better? 

For example, when adding numbers, I can't remember when the answer is
supposed to be positive, or when it's supposed to be negative.

Date: 10/27/2003 at 22:00:34
From: Doctor Ian
Subject: Re: algebra 

Hi Ivette,

In my opinion, that's the wrong question to be asking.  Memorizing
formulas is a poor solution in the long run, because you'll forget
everything that you don't use on a daily basis--especially when you're 
under a lot of stress, for example, during a test!

Whenever you find yourself trying to memorize something, that's a sign
that there's something you need to understand better.  

For example, when thinking about adding and subtracting positive and
negative numbers, keep in mind that they refer to opposite directions
on the number line.  So instead of '+this' and '-that', you can think
of them in terms of east (+) and west (-). 

(Or you can use north and south, or up and down, or in and out...
whichever pair of opposites seems most natural to you.) 

If I go 2 miles east, and 3 miles east, where do I end up?

  2 east + 3 east = 5 east            (2 + 3 = 5)

What if I go 2 miles east, and then 3 miles west?  

  2 east + 3 west = 1 west            (2 + -3 = -1)

How can I see this, without memorizing it?  Well, I can break the trip
west into two parts:

  2 east + 2 west + 1 west 

Now the first to cancel each other out!

  (2 east + 2 west) + 1 west = 0 + 1 west = 1 west 

Now, we _could_ do this just using signs, instead of directions:

  2 + (-3) = 2 + (-2 + -1)

           = (2 + -2) + -1

           = 0 + -1

           = -1

But the important thing is, we can see how it works without having to
memorize any rules or formulas.

Does this make sense?   

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/